Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444408 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,6,7,8)(9,10,12,17,14,18,13,15)(11,16,21,19,20,24,22,23);(2,5)(6,8)(9,10)(11,19)(12,15)(13,17)(14,18)(16,21)(20,23)(22,24);(10,15,18,17)(16,23,24,19);(1,3,4,7)(2,6,5,8)(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23);(11,20)(16,24)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213590, #72213751, #72339031, … |
| maximal normal subgroup of | #72213590, #72213751, #72339031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367247);